2.10 Universes and the univalence axiom

Given two types A and B, we may consider them as elements of some universe type 𝒰, and thereby form the identity type A=𝒰B.
As mentioned in the introduction, univalence is the identification of A=𝒰B with the type (A≃B) of equivalences from A to B, which we described in §2.4 (http://planetmath.org/24homotopiesandequivalences).
We perform this identification by way of the following canonical function.

Lemma 2.10.1.

For types A,B:U, there is a certain function,

𝗂𝖽𝗍𝗈𝖾𝗊𝗏:(A=𝒰B)→(A≃B),

(2.10.1)

defined in the proof.

Proof.

We could construct this directly by induction on equality, but the following description is more convenient.
Note that the identity function𝗂𝖽𝒰:𝒰→𝒰 may be regarded as a type family indexed by the universe 𝒰; it assigns to each type X:𝒰 the type X itself.
(When regarded as a fibration, its total space is the type ∑(A:𝒰)A of “pointed types”; see also §4.8 (http://planetmath.org/48theobjectclassifier).)
Thus, given a path p:A=𝒰B, we have a transport function p*:A→B.
We claim that p* is an equivalence.
But by induction, it suffices to assume that p is 𝗋𝖾𝖿𝗅A, in which case p*≡𝗂𝖽A, which is an equivalence by Example 2.4.7 (http://planetmath.org/24homotopiesandequivalences#Thmpreeg1).
Thus, we can define 𝗂𝖽𝗍𝗈𝖾𝗊𝗏⁢(p) to be p* (together with the above proof that it is an equivalence).
∎

We would like to say that 𝗂𝖽𝗍𝗈𝖾𝗊𝗏 is an equivalence.
However, as with 𝗁𝖺𝗉𝗉𝗅𝗒 for function types, the type theory described in http://planetmath.org/node/87533Chapter 1 is insufficient to guarantee this.
Thus, as we did for function extensionality, we formulate this property as an axiom: Voevodsky’s univalence axiom.

Axiom 2.10.3 (Univalence).

Technically, the univalence axiom is a statement about a particular universe type 𝒰.
If a universe 𝒰satisfies this axiom, we say that it is univalent.
Except when otherwise noted (e.g. in §4.9 (http://planetmath.org/49univalenceimpliesfunctionextensionality)) we will assume that all universes are univalent.

Remark 2.10.4.

It is important for the univalence axiom that we defined A≃B using a “good” version of isequiv as described in §2.4 (http://planetmath.org/24homotopiesandequivalences), rather than (say) as ∑(f:A→B)qinv⁢(f).

In particular, univalence means that equivalent types may be identified.
As we did in previous sections, it is useful to break this equivalence into: